Selfsimilar expanders of the harmonic map flow

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www.elsevier.com/locate/anihpc

Selfsimilar expanders of the harmonic map flow

Pierre Germain

^a

, Melanie Rupflin

^b,^∗^,1

aCourant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, NY, USA bWarwick Mathematics Institute, University of Warwick, Coventry, United Kingdom

Received 9 November 2010; accepted 6 June 2011 Available online 14 July 2011

Abstract

We study the existence, uniqueness, and stability of self-similar expanders of the harmonic map heat flow in equivariant settings.

We show that there exist selfsimilar solutions to any admissible initial data and that their uniqueness and stability properties are essentially determined by the energy-minimising properties of the so-called equator maps.

Résumé

On étudie l’existence, l’unicité et la stabilité de solutions auto-similaires issues d’une singularité, pour le flot gradient des applications harmoniques, dans le cadre équivariant. On montre l’existence de telles solutions auto-similaires, et comment leurs propriétés d’unicité et de stabilité sont étroitement reliées à la minimisation ou non de l’énergie de Dirichlet par l’application équateur.

1. Introduction

1.1. The harmonic map heat flow and its solutions

The harmonic map heat flow is defined as the negative gradient flow of the Dirichlet energy of maps between manifolds. For a mapu(x, t )fromR^d× [0,∞)to a manifoldN, which we see as embedded in some Euclidean space with second fundamental formΓ, this equation reads

∂tu−u=Γ (u)(∇u,∇u) onR^d× [0,∞), u(t=0)=u₀.

Choosingu₀∈H¹(finite energy data), Struwe [36], Chen [5], and Chen and Struwe [6] (see also Rubinstein, Sternberg and Keller [30]) were able to build up weak solutions. In the critical dimensiond=2 the question of uniqueness of weak solutions has been analysed by Freire [12], Topping [38], Bertsch, dal Passo and Van der Hout [1] and the

* Corresponding author. Tel.: +44 (0)2476150774.

E-mail address:M.Rupflin@warwick.ac.uk (M. Rupflin).

1 Partially supported by the Swiss National Science Foundation and The Leverhulme Trust.

doi:10.1016/j.anihpc.2011.06.004

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second author of this paper [31]. On the other hand, the question of uniqueness is still open in the supercritical dimensionsd3 that we consider here. On the one hand, examples of non-uniqueness have been obtained by Coron [8] and Hong [18]. On the other hand uniqueness can be obtained by working at the scaling of the equation: Koch and Lamm [21] proved local well-posedness for data which are close inL^∞to a uniformly continuous map; Wang [40]

obtained local well-posedness for data small enough in BMO; finally Lin and Wang [22] showed uniqueness in C([0, T], W^1,n).

1.2. Equivariant setting

We shall assume that the target manifold is spherically symmetric, more precisely that it admits coordinates (s, ω)∈R×Sⁿ⁻¹in which its metric readsds²+g²(s) dω²

Sⁿ⁻¹. We shall furthermore assume that the solution map is equivariant, namely in these coordinatesu(t, x)=(h(t,|x|), χ (_|^x_x_|)), whereχ is ak-eigenmap, see Section 2 for the details. Then the above equation reduces to a scalar one:

⎧⎨

⎩

h_t−h_rr−d−1 r h_r+ k

r² gg

(h)=0, h(t=0)=h₀.

The archetype of such a situation is given by corotational maps into thed dimensional sphere, in which caseg=sin, k=1,χ=I d, and the ansatz readsu(t, x)=(h(t,|x|),_|^x_x_|). The equator of the sphere corresponds to the solution h≡^π₂; it is a trivial solution of the harmonic map heat flow. In our more general equivariant framework, an equator of a rotationally symmetric manifold is a lateral sphere of N with locally maximal diameter; it corresponds to the constant in time solution of the harmonic map flow given byh≡s,sa local maximum ofg².

1.3. Obtained results: existence and uniqueness of self-similar solutions

We investigate the above equation with data of the typeh₀≡s∈R; the expected solutions are self-similar, i.e. of the type

h(x, t )=ψ |x|

√t .

We first establish (in Theorem 2.3) the existence of such a self-similar profile for anys. The next question is that of uniqueness; roughly speaking, we are able to prove the equivalence of the two following statements (see Theorems 2.2 and 2.3 for the details):

• For any givens, there exists a unique self-similar profile.

• The equator maph≡sminimises the Dirichlet energy on the unit ball among all functions in the same equivari- ance class and with prescribed valueh=son the boundary of the ball.

This equivalence stated above can be established either by ODE, or variational methods; we follow both paths, which yield complementary results. We would like to mention that parts of the above result were known to Angenent, Ilmanen and Velazquez (unpublished work, announced in [19]). Also, Biernat and Bizon [2] obtained numerical and analytical results for the above problem.

1.4. Implications for the uniqueness of solutions to the Cauchy problem

The self-similar solutions we consider are (locally in space) of finite energy; actually, they barely miss the conditions for which uniqueness or local well-posedness was stated above, thus proving the optimality of our results.

Another non-uniqueness result for the harmonic map flow from R³ to the sphere is due to Coron [8], see also Hong [18]. These arguments are more indirect and lead to only two (genuinely) different solutions, as opposed to ours, which yield a precise description of the non-unique solutions and a large number of genuinely different solutions.

Though Coron’s approach is very different from ours, both, interestingly enough, rely on the energy-minimising properties of certain harmonic maps.

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Lastly, incoming self-similar solutionsu(x, t )=v(√^x

−t),t <0 have drawn a lot of attention, since they provide instances of singularity formation, or blow up, from smooth data: see Ilmanen [19], Fan [11] and Gastel [15]. A com- bination of their results and ours yields, in some cases, non-unique continuations after the blow up time. Biernat and Bizon [2] studied the question of continuation if the blow up forms along a certain profile which is numerically stable;

they gathered analytic and numerical evidence for unique continuation in that case.

1.5. Related results: wave maps and nonlinear heat equation

A result similar to the one above is known for the wave map equation: see Shatah [33], Cazenave, Shatah and Tahvildar Zadeh [4], and the first author of the present article [16].

The equivalence stated above is also reminiscent of the situation for the nonlinear heat equation with power non- linearity:

∂_tv−v= |v|^αv onR^d× [0,∞), v(t=0)=v0.

In the supercritical range, i.e. forα >_d₋⁴₂, the equation with self-similar datav0=_|_x_|α yields self-similar solutions v(t, x)=t⁻^1/αψ (√^r

t). With respect to the issue of uniqueness it turns out that there are deep analogies between this equation and the harmonic map heat flow: the analog of the equator map is the stationary solution _|_x^β_|_2/α, withβ= (_α²(d−2−²_d))^1/α, and it is stable if and only if self-similar solutions are unique: this is the case ifα > ⁴

d−4−2√ d−1. For this and related results we refer to [17,41,26,10,13,35,25,24].

1.6. Obtained results: stability of self-similar solutions

In Theorems 2.5, 2.6, and 2.7, we examine the stability of our self-similar solutions, with respect to small perturbations of the data at time 0, and at time 1. We are not able to give a complete picture, but we can characterise to a large extent stable and unstable settings. The methods employed are spectral (in particular the analysis of Sturm–Liouville problems) for the linearised problem, and resort to nonlinear analysis for the full equation.

2. Statement of the results

2.1. The problem under study

We consider selfsimilar weak solutions of the harmonic map heat flow

∂_tu−u=Γ (u)(∇u,∇u) onR^d× [0,∞) (1)

from Euclidean spaceR^dinto a smooth target manifoldN. We focus here on expanding selfsimilar solutions

u(x, t )=v x

√t , t >0, x∈R^d

for a suitable mapv:R^d→N. By the translation invariance of (1) these maps represent all solutions of (1) which are selfsimilar in forward time-direction up to translations in space-time. Such solutions in the natural energy-space

(u, u_t)∈L^∞_loc

[0,∞)H˙_loc¹ R^d

×L²_loc

R^d× [0,∞)

(2) of (1) exist only in supercritical dimensionsd3.

These self-similar maps correspond to data which are homogeneous of degree 0 u(t=0)(x)=u₀

x

|x| , x∈R^d\ {0}. (3)

Our aim in the present article will be to understand the existence, uniqueness, and stability properties of the Cauchy problem for (1) with homogeneous initial data.

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2.2. Geometric setting

We consider maps from a fixed Euclidean spaceR^d,d3, into a smooth rotationally symmetric target manifold Nⁿwithout boundary. We introduce coordinates(s, ω)∈R×Sⁿ⁻¹onNin which the metric is given by

ds²+g²(s) dω²_n₋₁.

Heredω²_n₋₁denotes the standard metric of the sphereSⁿ⁻¹andgshall be a smooth function, symmetric with respect to each pointpwhereg(p)=0. For these special valuespof the lateral coordinate which represent the poles ofN, it is necessary to assume that|g(p)| =1 in order to obtain a smooth manifold. The coordinatesand the functiongare of course periodic ifN is compact.

Observe that the (intrinsic) diameter of thelateral sphere C_s:=

(s, ω): ω∈Sⁿ⁻¹

, s∈R

is equal toπ|g(s)|. We therefore callCs anequatorofN ifs is a local maximum ofg². Similarly, we call a lateral sphere whose diameter is locally minimal but positive aminimal sphere.

We consider for the moment both compact and non-compact target manifoldsN, but we want to assume throughout this work that

sup

s∈R

d² ds²

g² (s)

+ g²(s)

1+s²<∞. (4)

For simplicity, we also exclude targets for which g has roots with multiplicity greater than one or for which the functions→_ds^d²2(g²)(s)is constant on an interval of positive length.

We consider maps fromR^dtoNwith the following type of symmetry.

Definition 1.Letd, n∈N.

(i) We call a mapχ:S^d⁻¹→Sⁿ⁻¹a (k-)eigenmap, ifχ is an eigenfunction of the negative Laplacian−_Sd−1 with constant energy density

|∇χ|²=k.

(ii) Let Nⁿ be a rotationally symmetric manifold and let χ:S^d⁻¹→Sⁿ⁻¹ be an eigenmap. We say that a map u:R^d→Nⁿisχ-equivariantif there exists a functionh:[0,∞)→Rsuch that

u(x)=Rχh(x):=

h

|x| , χ

x

|x|

with respect to the rotationally symmetric coordinates introduced onN.

Eq. (1) becomes in equivariant coordinates h_t−h_rr−d−1

r h_r+ k

r²G(h)=0 (5)

(whereG:=gg), see Lemma 3.1. In particular forh≡s∈Rthe mapR_χhis harmonic and thus a trivial solution of the harmonic map flow if and only ifG(s)=0, i.e. ifC_s is either a pole, a minimal sphere or an equator.

Let us remark that the spectrum of the negative Laplacian on the sphereS^d⁻¹ l(d−2+l): l∈N

contains no eigenvalues smaller thand−1. An example of a(d−1)-eigenmap is of course the identityid:S^d⁻¹→ S^d⁻¹with the corresponding equivariant maps being the corotational mapsx→(h(|x|),_|^x_x_|). The components of gen- eral eigenmaps with eigenvalueλ_l=l(d−2+l)are given by the restriction ofl-homogeneous, harmonic polynomials to the sphere, see [9, Chapter VIII].

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2.3. Equator maps and their minimising properties

Given any equatorC_sofNand any eigenmapχ, we define the correspondingequator mapby u=u_{χ ,s}:=R_χh

for the constant functionh≡s. Note that this equator map and its properties depend both on the eigenmapχ and on the value ofs.

Definition 2.LetC_s be an equator of a rotationally symmetric manifoldN and letχ be an eigenmap. We say that the equator mapu_{χ ,s} isχ-energy-minimisingif it minimises the Dirichlet energy

E

u, B₁(0)

=1 2

B1(0)

|∇u|²dx

in the set Fχ ,s:=

R_χh: h:[0,1] →Rwithh(1)=s ofχ-equivariant functions with the same boundary data.

Notice that we do not demand that the equator mapu_{χ ,s}be energy-minimising in the larger class of maps F =

u∈H¹(B₁, N ): u|∂B1=u_{χ ,s}

∂B₁

.

We cannot exclude the possibility of symmetry breaking in the sense that

vinf∈FE(v, B₁) < inf

v∈Fχ ,s

E(v, B₁).

An example for such an occurrence in a related context ofG-equivariant harmonic map was given by Gastel [14]

based on the analysis of singularities by Brezis, Coron and Lieb [3].

The following proposition provides a simple criterion to test whether or not a given equator map is χ-energy- minimising.

Proposition 2.1.Letd3, letNⁿbe a smooth, rotationally symmetric manifold and let χ:S^d⁻¹→Sⁿ⁻¹be a k- eigenmap. LetCsbe an equator ofNand recallG:=g·g.

(i) If

−4kG s

< (d−2)², (6)

the equator map islocally (i.e. for small perturbations)χ-energy-minimising.

(ii) If

−4kG s

> (d−2)², (7)

the equator mapu_{χ ,s}is not(locally)χ-energy-minimising.

(iii) Suppose that

−4kG(s)(d−2)² fors∈

s−S, s+S

whereS:=_d²^√₋^k₂· g∞. Thenu_{χ ,s} isgloballyχ-energy-minimising.

Applying the above criterion to the case where the target manifold is the sphere S^d with the standard metric ds²+sin(s)²dω²_d₋₁and the maps are corotational, χ=I d, gives the well-known result: the equator mapR_χ^π₂ is energy-minimising if and only ifd7.

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2.4. Existence and uniqueness results

We are able to prove existence of solutions to (1) for homogeneous data in essentially all cases; notice that we are dealing with infinite energy solutions, thus the existence theorems by Chen [5] and Chen and Struwe [6] do not apply here. The question of uniqueness is much more interesting; roughly speaking, we shall prove that solutions to the Cauchy problem for (1) with homogeneous data are unique if and only if the equator map is energy-minimising.

More precise formulations of this idea are contained in the two following theorems.

We begin with a very general setting(N, χ ), where the equator map is notχ-energy-minimising.

Theorem 2.2.Letd3, letNⁿbe a rotationally symmetric manifold such that(4)is satisfied and letχ:S^d⁻¹→Sⁿ⁻¹ be a fixed eigenmap. Assume thatN has an equator mapu_{χ ,s} which is notχ-energy-minimising.

Then there exists a selfsimilar andχ-equivariant weak solutionu∈H_loc¹ (R^d× [0,∞))of the initial value problem (1),(3)that is not constant in time for the initial datau₀=u_{χ ,s}.

In the statement of our uniqueness result we will impose the following restrictions on the functiongrepresenting the metric ofN.

Condition (C1).LetC_s be an equator of a compact, rotationally symmetric manifoldNand lets₁< s< s₂be the local minima ofg²to the left and to the right ofs, i.e. the local minima ofg²such thatg²|[s₁,s]is increasing while g²|[s,s₂]is decreasing.

We then demand that G

s

= min

s∈[s₁,s₂]G(s) (recallG(s):=g(s)g(s)).

For manifolds that contain a minimal sphereC_s₀we furthermore impose

Condition (C2).Letkbe any given eigenvalue of−_Sd−1. We say that a rotationally symmetric manifoldN fulfils condition (C2) (fork) if for each minimal sphereC_s₀ ofN

G(s₀)d−1 k .

Conditions (C1) and (C2) are fulfilled for a wide variety of rotationally symmetric manifolds, in particular for round spheres and for rotationally symmetric ellipsoids.

Theorem 2.3. Let d 3, let Nⁿ be a compact, rotationally symmetric manifold and let χ:S^d⁻¹→Sⁿ⁻¹ be an eigenmap.

(i) There exists a selfsimilar and equivariant weak solution of (1)for any admissible initial data, i.e. for every map u₀(x)=(s, χ (_|^x_x_|)),s∈R.

(ii) Assume that all equator maps of the manifoldNareχ-energy minimising and that conditions(C1)and(C2)are satisfied. Then the solution of (i)is unique in the class of all equivariant and selfsimilar weak solutions.

(iii) Assume that the manifoldNhas an equatorC_s such that

−4kG s

> (d−2)²,

i.e. such that the corresponding equator map is not even locally energy minimising. Then given any number K∈N, there exists a neighbourhoodUKofssuch that the initial value problem(1),(3)has at leastKdifferent weak solutions which areχ-equivariant and selfsimilar for each initial datau₀(x)=(s,_|^x_x_|)withs∈U_K.

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Remark 2.4.

1. All solutions obtained in Theorems 2.2 and 2.3 satisfy the monotonicity formula of Struwe [37]. For (constant in time) equivariant harmonic maps this follows since the maps are stationary harmonic (see Lin and Wang [23, Lemma 7.4.1]). For more general selfsimilar solutions the monotonicity formula can be shown using the asymptotics of solutions of (14) of Lemma 5.1 below.

2. We can replace assumption (C1) in statement (ii) by demanding that−4kG(s)(d−2)²for everys∈Rwhich is a weaker assumption in view of Proposition 2.1.

2.5. Stability at timet=1

In the previous section, we characterised precisely the existence and uniqueness properties of self-similar solutions to the harmonic map heat flow. Our aim now is to study their stability: we focus in this section on the effect of a perturbation at timet=1, and in the next on a perturbation occurring at timet=0.

Letψ be one of the self-similar profiles whose existence has been established, and consider a perturbationu= R_χ[ψ (√^·

t)+f]. For data given att=1, the Cauchy problem becomes

⎧⎨

⎩

f_t−f_rr−d−1 r f_r+ k

r²

G

f +ψ √·

t

−G

ψ √·

t

=0, f (t=1)=f0.

(8) By scaling invariance, it is of course equivalent to study the problem fromt=1 or any other positive time.

Suppose first that ψ is provided by Theorem 2.2. In the proof given in Section 4 we will construct ψ as the minimiser of the functional

E(f ):=

∞ 0

f²+ k r²

g²(f )−g²

s

r^d⁻¹e^r²^/4dr,

which is very reminiscent of the well-known monotonicity formula for the harmonic map heat flow [37]. Thus our first stability result essentially corresponds to a forward time version of the monotonicity formula.

Theorem 2.5.Letψbe given by Theorem2.2, and letf solve(8). ThenE(f (√

t·)+ψ )is a decreasing function of time.

Even thoughEis only minimised atψ, it is not clear to us to what extentE(f (√

t·)+ψ )controlsf.

Consider now a general profileψ, given by Theorem 2.3; we want to investigate linear stability. The linearised version of (8) reads

⎧⎨

⎩

r²G

ψ √·

t

f=0, f (t=1)=f0.

(9) A spectral analysis of the above problem in self-similar variables will lead to the following result.

Theorem 2.6.Letψbe given by Theorem2.3, and letvbe a solution of (9).

(i) Ifψis monotone, then v(t )

L²(e^r

2

4tr^d⁻¹dr)t^d⁴⁻¹v₀

L²(e^r

2 4r^d−1dr)

. (10)

In particular ford4the functiont→ v(t )

L²(e^r

2 4tr^d−1dr)

is decreasing.

(ii) IfψhasKlocal extrema, there existsγ >0andKlinearly independent initial datav₀such that v

L²(e^r

2

4tr^d−1dr)t^γ⁺^d⁴⁻¹.

In particular ford4this corresponds to a growing norm.

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Estimate (10) is optimal; in fact, it is satisfied with equality for the solutionv(t ):=_2t¹ ·^√^x_tψ(√^x

t)of (9) which we obtain by differentiating (5) in time.

2.6. Stability at timet=0

We focus in this section on the effect of a perturbation at timet=0.

We start with the most simple type of self-similar solutions: the maps which are constant in time, mappingR^donto an equator or a minimal sphere. Ifu=R_χ(f +s), the equation under study is

⎧⎨

⎩

r² G

f +s

−G s

=0, f (t=0)=f₀.

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Theorem 2.7.Letd3, letNⁿbe a rotationally symmetric manifold such that(4)is satisfied and letχ:S^d⁻¹→Sⁿ⁻¹ be a fixed eigenmap. Supposes is such thatG(s)=0;it corresponds to a constant solution of (1)given byu_{χ ,s}. Consider the perturbed equation(11).

(i) If

kG s

>−(d−2)²

4 ,

this equation is linearly stable(in more precise terms:the Cauchy problem associated with the linear part of Eq.(11)is globally well-posed inL², and theL²norm is decreasing).

(ii) If

kinf

R G>−(d−2)²

4 ,

there exists a global weak solutionf to the above equation, satisfying

f²_L∞([0,∞),L²(R^d))+ ∇f²_L2([0,∞),L²(R^d))f0²2. (12) (iii) If

G s

>0,

i.e. ifs is the coordinate of a pole or a minimal sphere, the above equation is globally well-posed inL^∞for small data(in more precise terms:iff0is small inL^∞, there exists a solutionf of (11)inL^∞([0,∞), L^∞), which is unique in a small enough ball and depends continuously onf₀).

Remark 2.8.

1. The weak solutions in (ii) do not share – even locally – the same functional setup as the Struwe solutions: whereas the former givef ∈L^∞_t L², the latter would roughly correspond tof∈L^∞_t H˙¹.

2. Notice that the spaces (for the data as well as the solution) for which well-posedness is proved in (ii) and (iii) are at the same scaling as the equation: their norms are invariant by the scaling which leaves the equation invariant. We do not claim any optimality for these spaces: there should exist larger spaces in which the equation is well-posed.

3. The above theorem gives sufficient conditions onG(s)for various kinds of stability results to hold true. We ask in Section 7.4 whether they are also necessary. The answer is shown to be yes for (i) and (ii).

For non-constant selfsimilar solutions we obtain the following stability result

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Theorem 2.9.In the setting which was just described, consider the perturbed equation for a self-similar profile:If u(t )=R_χ[ψ (√^·

t)+f]solves(1)thenf solves

⎧⎨

⎩

r²

G

f +ψ √·

t

−G

ψ √·

t

=0, f (t=0)=f0.

Suppose thatψis such thatG(ψ (r)) >0for allr >0. Then the above equation is globally well-posed inL^∞, namely, forf₀small enough inL^∞, there exists a solutionuinL^∞_t L^∞, which is unique in a small enough ball, and depends continuously on the data.

This theorem follows by the same arguments as Theorem 2.7(iii), thus we skip its proof.

Notation.The notationABmeans: there exists a constantCsuch thatACB. 3. Preliminaries

3.1. Weak solutions of the harmonic map flow in the equivariant setting

LetNⁿ be a rotationally symmetric manifold, letg:R→Rbe the function describing the metric ofN and let χ:S^d⁻¹→Sⁿ⁻¹ be an eigenmap to eigenvalue k∈N. A short calculation shows that the Dirichlet energy of an equivariant mapv=R_χhis given by

E

v, BR(0)

=1 2

B_R(0)

|∇v|²dx=c_d 2

R 0

h²+ k r²g²(h)

r^d⁻¹dr

forc_d= |S^d⁻¹|denoting the Hausdorff-measure of thed−1 dimensional unit sphere.

In view of condition (4) the set of functionsh which induce equivariant maps with locally finite energy can be described by

Definition 3.Givend∈Nand a ballB_R=B_R(0)⊂R^d we define H_rad¹ (B_R):=

h:[0, R] →R:

R 0

h²+h²

r² r^d⁻¹dr <∞

,

and set H_rad¹

R^d :=

R>0

H_rad¹ (B_R).

Observe that the equivariant functionR_χh:R^d→N is an element ofH_loc¹ (R^d)but not necessarily ofH¹(R^d)if h∈H_rad¹ (R^d). Let us also remark that the global energiesE(u(t ),R^d)of solutions of the harmonic map heat flow (1) are in general infinite.

Direct computations (see e.g. [16]) lead to the following characterisation of equivariant weak solutions of the harmonic map heat flow.

Lemma 3.1.Consider a rotationally symmetric manifoldNⁿwith metric described byg∈C¹(R)and letχ:S^d⁻¹→ Sⁿ⁻¹be ak-eigenmap.

(i) Letube an element of the energy-space(2)of the formu=R_χhfor a functionh:R⁺₀ ×R⁺₀ →R. Thenuis a weak solution of (1)if and only ifhsolves the scalar partial differential equation

(10)

h_t−

h_rr+d−1 r h_r− k

r²G(h)

=0 onR⁺₀ ×R⁺₀ (13)

in the sense of distributions.

(ii) Letube an element of the energy-space(2)that is of the formu(x, t )=R_χh(√^x

t),t >0, for someh:R⁺₀ →R. Thenuis a weak solution of (1)if and only ifhsolves the differential equation

h+ d−1

r +r

2 h− k

r²G(h)=0 on(0,∞). (14)

Remark that we can rewrite Eq. (14) in divergence-form as d

dr

r^d⁻¹e^r²^/4h(r)

=kr^d⁻³e^r²^/4G(h). (15)

It can be easily checked that a selfsimilar mapu(x, t )=R_χh(√^x

t)is an element of the energy-space if and only if h∈H_rad¹ (R^d)and

1 0

(√ t )^d⁻⁴

R/√

t 1

h²r^d⁺¹dr dt <∞. (16)

At first glance the assumptionh∈H_rad¹ (R^d)imposes only a mild constraint on the behaviour ofhnearr=0 while the condition (16) seems to seriously restrict the allowed behaviour at infinity. We will see later that the converse is true for solutions of Eq. (14). Indeed, the first derivative of each solution of (14) decays sufficiently fast for (16) to be fulfilled, but most solutions of (14) blow up asr→0 in such a way thath /∈H_rad¹ (R^d).

Let us finally remark that the trace of a selfsimilar mapu(x, t )=R_χh(√^x

t)on the time sliceR^d× {0}is given by u(x,0)=(s, χ (_|_x^x_|))ifhconverges tos∈Rasr→ ∞.

3.2. Characterisation of energy-minimising equator maps

As remarked in [16], the criterion given in Proposition 2.1 is closely related to the value of the optimal constant in the Hardy inequality.

Lemma 3.2.Letd3. ThenC_H=_(d₋⁴₂₎2 is the optimal constant such that the Hardy inequality 1

0

w²r^d⁻³drCH

1 0

w²r^d⁻¹dr (17)

holds true for allw∈H_rad¹ (B₁)withw(1)=0.

For a proof of this result we refer to [39].

Proof of Proposition 2.1. The proofs of (i) and (ii) follow directly from Hardy’s inequality, as can be seen in [16].

Let us therefore assume that kG(s)−(d−2)²

4 = −C_H⁻¹ for alls∈

s−S, s+S whereS:=√

kCH · g∞= ²_d^√₋^k₂· g∞. Using the quadratic Taylor expansion ofg²arounds, we find for these values ofs

k

g²(s)−g² s

−C_H⁻¹ s−s2

.

Remark that this estimate is trivially true if|s−s|S.

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The above Hardy inequality thus implies that for allh∈H_rad¹ (B₁)withh(1)=s 1

0

h²+ k r²

g²(h)−g²

s

r^d⁻¹dr 1

0

h−s²−C⁻_H¹ r²

h−s2

r^d⁻¹dr0 and thus thatE(R_χh B₁)E(u, B₁). 2

4. The variational approach: proof of Theorem 2.2

We prove the first non-uniqueness result, Theorem 2.2 by variational methods. Contrary to the arguments used for the proof of Theorem 2.3, we do not require any restrictions on the manifoldNother than the general assumption (4).

Theorem 2.2 is thus valid also for a large class of non-compact rotationally symmetric target manifolds.

4.1. The variational problem

LetN be a rotationally symmetric manifold,χ an eigenmap and let Cs be an equator of N. Assume that the equator mapu_{χ ,s} is notχ-energy-minimising. According to the discussion in Section 3.1 we need to establish the existence of a non-constant solutionh∈H_rad¹ (R^d)to Eq. (14) with limr→∞h(r)=swhich satisfies condition (16).

We consider the set F:=

f ∈H_rad¹ R^d

: supp(f )⊂⊂ [0,∞) and take its closureFwith respect to the norm

f²:= f²+|f|²

r² r^d⁻¹e^r²^/4dr.

Let us remark that condition (16) is trivially fulfilled for elements ofFand that functions inFconverge to zero as r→ ∞. In view of the divergence form (15) of Eq. (14) we consider the variational integral

E(f ):=

∞ 0

f²+ k r²

g² s+f

−g²

s

r^d⁻¹e^r²^/4dr (18)

on the reflexive space(F, · )(Eis finite onF sinceg(s)=0). We prove that this functional has the following properties

1. E(·)is weakly lower semi-continuous and bounded from below on(F, · ).

2. If the equator mapu_s,χ is notχ-energy-minimising, then inf

f∈FE(f ) <0=E(0).

We therefore find thatEachieves its global minimum for a functionf∈Fthat is not identically zero. Consequently s+f is a non-constant solution of (14) that induces a selfsimilar weak solution of the harmonic map flow for initial datau₀=u_{χ ,s} different from the time-independent equator map.

It remains to prove the above claims aboutE.

4.2. Proof of claim 1

We use thatC₁:=sup_s_∈R|_ds^d²2g²(s)|<∞by assumption (4) and estimate g²

s+f

−g² s

−min

C₁f², g² s

. Given anyR >0 and anyf∈F, we thus obtain

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E(f )= ∞

0

f²+ k r²

g² s+f

−g²(s)

r^d⁻¹e^r²^/4dr

∞

0

f²r^d⁻¹e^r²^/4dr−kC1

∞ R

f²r^d⁻³e^r²^/4dr−kg² s^R

0

r^d⁻³e^r²^/4dr

∞

0

f²r^d⁻¹e^r²^/4dr−CR⁻² ∞ R

f²r^d⁻¹e^r²^/4dr−C(R)

for a constantC(R)independent off.

In the weighted spaceFthe Hardy inequality ∞

0

f² 1+r²

r^d⁻³e^r²^/4drC ∞

0

f²r^d⁻¹e^r²^/4dr (19) holds true for a universal constantC=C(d), see e.g. [39]. Choosing the numberR >0 in the above estimate large enough, we thus obtain a uniform lower bound forEonF.

Remark that inequality (19) shows furthermore that an equivalent norm to · on F is given by |||f|||²:=

|f|²r^d⁻¹e^r²^/4dr. The weak lower semi-continuity ofEthen follows from the estimate ∞

R

g² s+f

−g² s

r^d⁻³e^r²^/8C1

R²f² and the lemma of Fatou applied on finite intervals[0, R].

4.3. Proof of claim 2

In order to prove the second property ofE, we define a family of weighted energies(Eλ)λ∈[0,1]on the spaceFby E_λ(f ):=

∞ 0

f²+ k r²

g² s+f

−g²

s

r^d⁻¹e^λr²^/4dr.

Note the scaling

E_λ(h)=λ⁻^d−2² ·E₁

h ·

√λ

=λ⁻^d⁻²² ·E

h ·

√λ

.

Since the equator mapu=u_s,χ is by assumption not energy-minimising, there exists a functionh∈H_rad¹ (B₁)with h(1)=0 and

E Rχ

s+h , B1

−E u, B1

=c_d 2

1 0

h²+ k r²

g² s+h

−g²

s

r^d⁻¹dr <0.

Extendinghby zero on[1,∞), we thus obtain thatE₀(h) <0 and by continuity ofλ→E_λ(h)alsoE_λ(h) <0 for λ >0 small. Consequently

inf

f∈FE(f )=λ^d−2² inf

f∈FE_λ(f ) <0 as claimed.

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5. Properties of the associated ordinary differential equation

5.1. Existence, uniqueness, and asymptotic behaviour

In this section we collect several important properties of solutions to the differential equation (14) characterising selfsimilar solutions in the equivariant setting. We shall assume from now on thatNis compact and thus in particular thatg,gandgare bounded periodic functions onR.

We first show that the behaviour of arbitrary solutionshof (14) forr→ ∞can be described by Lemma 5.1.

(i) Lethbe any solution of (14). Then there exists a constantC=C(h)such that h(r)C

r³ for allr1.

(ii) This inequality holds true with a universal constant C =C(g, k) for all solutions h of (14) satisfying limr→0r·h(r)=0.

Proof. The quantity

V (r)=V (h)(r):=r²h(r)²−kg² h(r)

(20) is decreasing for any non-constant solutionhof (14) with

V(r)= −r²h(r)²

2(d−2)

r +r

. (21)

The possible behaviour ofF (r):=V (r)+kg²(h(r))=r²|h(r)|²is thus constrained by F(r)+rF (r)2kG(h)·h(r)r

2F (r)+C r³. Integrating the above inequality we then obtain that

F (r)

e^1/4F (1)+C

·e⁻^r²^/4+C

r⁴ (22)

which leads to the desired estimate of statement (i). This estimate is independent of the solutionhifrh(r)→0 asr→ 0 since in this case 0V (0)V (r)F (r)−kg²_∞for everyr >0, and thus in particular|F (1)|kg²_∞. 2

An important consequence of Lemma 5.1 is that each solution h of (14) converges as r→ ∞ in such a way that condition (16) is satisfied. In order to find selfsimilar solutions of the harmonic map heat flow we can therefore concentrate on finding solutions of (14) that are elements ofH_rad¹ (R^d).

Proposition 5.2. Lets₀ be a local minimum ofg² and leta∈R. Then there exists a solutionh_a∈C²((0,∞))∩ C⁰([0,∞))of Eq.(14)such that

h_a(0)=s₀ and lim

r→0r⁻^γ

h_a(r)−s₀

=a, (23)

whereγ=¹₂(

(d−2)²+4kG(s0)−(d−2)). Additionally,r¹⁻^γh_a(r)→γ aasr→0, and thusha∈H_rad¹ (R^d).

Assuming furthermore that condition(C2)is satisfied, this solution is uniquely determined by(23).

Let us remark that the solutions (h_a)of (14) constructed in Proposition 5.2 induce a one-parameter family of selfsimilar weak solutions of the harmonic map flow. In fact, as we will prove in Section 6, the only other solutions of (14) which induce selfsimilar weak solutions of (1) are the constant functionsh=s, forC_s an equator ofN.

This proposition can be obtained by well-known methods in the theory of ordinary differential equations and is presented in detail in [32, Appendix B.1]. Assumption (C2) is necessary only for the proof of the uniqueness aspect